Features

John Wallis sits at his desk in Oxford, England, surrounded by books and mathematical instruments. It is 1655. At the top of a piece of paper he writes the words “Proposition 190” again, hovers over the area underneath, and stops. Out of habit, he crosses the words out. At this point he stands up and sighs, looks out the window, and thinks about his simple, code-breaking youth, before Parliament “honored” him with this Professorship and Chair of Geometry—it really is a terribly drafty workspace. He wonders if these three years of blank pages could have been spent any differently.

At this point, Wallis considers the fact that he has done enough work for the day. He mulls over the idea of walking a few buildings away to talk with his good friend Seth Ward, the Professor of Astronomy, about stars or the rings of Saturn. Nice, healthy, real world stuff. Once again he writes “Proposition 190,” at the top of his sheet of paper. Then he tosses the thing aside to go have a visit across the lawn.

Wallis is writing a book called The Arithmetic of Infinitesimals. Years later, his modern translator will comment that it was “perhaps the one real stroke of genius in Wallis’ long mathematical career,” though she also called it an “often tedious approach through scores of uninspiring Propositions and Corollaries.” When it is done, it will do things like annoy the printer, who will be forced to spend three years fitting Wallis’ odd symbols and drawings into the unwieldy press. It will give Thomas Hobbes, philosopher, fits of cranky rage. It will bridge the gap between geometry and algebra. It will convince a 22-year-old Isaac Newton, after he finishes reading, to invent his calculus. Mostly, it will provide a formula for a funny number called pi, discovering an estimate that is better and cleaner than anything that has come before. After it is done, mathematician Doctor William Oughtred will proclaim Wallis’ “understanding and genius, who have not only gone, but also opened a way into those profoundest mysteries of art, unknown and not thought of by the ancients.”

Right now, however, Wallis is stuck at an uninspired part. In the Comment attached to the end of Proposition 189, he writes, “But here, at last, I am at a loss for words.” He whines at the cruel mathematical fates: “Until now we seem to have carried the thing through happily enough.” He is entirely lost, “For I do not see in what manner I may produce the quantity o.” This value represented by this little square, whose discovery represents the climax of the book, is really pi (If we take 4 and then divide it by o, we get π).

Pi is a number that appears in every circle: it is the ratio between the circumference and the diameter. In decimal form, it is a number that never ends and never repeats, following no discernible pattern. It is interesting and important because it shows up in all branches of mathematics, from probability to real analysis. The problem is that it’s impossible to define with an exact decimal value, because such a decimal would never end. Indeed, the history of mathematics can in some ways be defined by this side-pursuit, this Holy Grail quest for a good formula for pi. Wallis was one of the first to find one, and his formula laid the groundwork for those that came later.

At Proposition 190, Wallis is still groping for the formula. He doesn’t know all the details yet, though he suspects that he’s approaching something magnificent. Indeed, he calls this elusive o a “slippery Proteus whom we have in hand, both here and above, frequently escaping and disappointing hope.” At times like this the Arithmetica sounds more like a personal journal than a textbook, which is the way Wallis likes to write about math, letting his exuberance for the material pour out unobstructed. When he includes, in the autobiography he wrote near the end of his life, the requisite information about wife and children, he does so only dutifully: “On March 4. 1644/5. I married Susanna daughter of John and Rachel Glyde of Northjam in Sussex; born there about the end of January 1621/2 and baptized Feb. 3 following. By whom I have (beside other children who died young) a Son and two Daughters now surviving.”

But when he talks about his first exposure to mathematics his tone changes completely. “One evening as we were sitting down to supper,” he writes, “a Chaplain of Sir William Waller shewed me an intercepted Letter written in Cipher.” It was a curious little puzzle, and it suited Wallis’ interests from the start. “It was about ten a clock when we rose from Supper,” he writes. “I then withdrew to my chamber to consider of it . . . In about 2 hours time (before I went to bed) I had deciphered it.”

It was the first time he put mathematics to use. Educated at Emmanuel College in Cambridge, he had studied Latin, Greek, Hebrew, Theology, and Logic before undergoing the Holy Orders. As an afterthought, he taught himself rudimentary mathematics from his younger brother’s trade books over the Christmas holiday. With that basic knowledge, Wallis would develop some limited renown for his skill in decoding, working for whatever political group was in power at the moment, by using arithmetic and the laws of numbers to translate letters of utmost state importance.